Samuel Zambrano

Combinatorial detection of determinism in noisy time series

This paper deals with the distinction between white noise and deterministic chaos in multivariate noisy time series. Our method is combinatorial in the sense that it is based on the properties of topological permutation entropy, and it becomes especially interesting when the noise is so high that the standard denoising techniques fail, so a detection of determinism is the most one can hope for. It proceeds by i) counting the number of the so-called ordinal patterns in independent samples of length L from the data sequence and ii) performing a χ2 test based on the results of i), the null hypothesis being that the data are white noise. Holds the null hypothesis, so should all possible ordinal patterns of a given length be visible and evenly distributed over sufficiently many samples, contrarily to what happens in the case of noisy deterministic data. We present numerical evidence in two dimensions for the efficiency of this method. A brief comparison with two common tests for independence, namely, the calculation of the autocorrelation function and the BDS algorithm, is also performed.

NF-κB oscillations translate into functionally related patterns of gene expression

Several transcription factors (TFs) oscillate, periodically relocating between the cytoplasm and the nucleus. NF-κB, which plays key roles in inflammation and cancer, displays oscillations whose biological advantage remains unclear. Recent work indicated that NF-κB displays sustained oscillations that can be entrained, that is, reach a persistent synchronized state through small periodic perturbations. We show here that for our GFP-p65 knock-in cells NF-κB behaves as a damped oscillator able to synchronize to a variety of periodic external perturbations with no memory. We imposed synchronous dynamics to prove that transcription of NF-κB-controlled genes also oscillates, but mature transcript levels follow three distinct patterns. Two sets of transcripts accumulate fast or slowly, respectively. Another set, comprising chemokine and chemokine receptor mRNAs, oscillates and resets at each new stimulus, with no memory of the past. We propose that TF oscillatory dynamics is a means of segmenting time to provide renewing opportunity windows for decision. DOI: http://dx.doi.org/10.7554/eLife.09100.001

DETECTING DETERMINISM IN TIME SERIES WITH ORDINAL PATTERNS: A COMPARATIVE STUDY

Detecting determinism in univariate and multivariate time series is difficult if the underlying process is nonlinear, and the noise level is high. In a previous paper, the authors proposed a method based on observable ordinal patterns. This method exploits the robustness of admissible ordinal patterns against observational noise, and the super-exponential growth of forbidden ordinal patterns with the length of the patterns. The new method compared favorably to the Brock–Dechert–Scheinkman independence test when applied to time series projected from the Henon attractor and contaminated with Gaussian noise of different variances. In this paper, we extend this comparison to higher fractal dimensions by using noisy orbits on the attractors of the Lorenz map, and the time-delayed Henon map. Finally, we make an analysis that enlightens the robustness of admissible ordinal patterns in the presence of observational noise.

Phase control of excitable systems

Here we study how to control the dynamics of excitable systems by using the phase control technique. Excitable systems are relevant in neuronal dynamics and therefore this method might have important applications. We use the periodically driven FitzHugh–Nagumo (FHN) model, which displays both spiking and non-spiking behaviours in chaotic or periodic regimes. The phase control technique consists of applying a harmonic perturbation with a suitable phase that we adjust in search of different behaviours of the FHN dynamics. We compare our numerical results with experimental measurements performed on an electronic circuit and find good agreement between them. This method might be useful for a better understanding of excitable systems and different phenomena in neuronal dynamics.

High-Throughput Analysis of NF-κB Dynamics in Single Cells Reveals Basal Nuclear Localization of NF-κB and Spontaneous Activation of Oscillations

NF-κB is a transcription factor that upon activation undergoes cycles of cytoplasmic-to-nuclear and nuclear-to-cytoplasmic transport, giving rise to so called “oscillations”. In turn, oscillations tune the transcriptional output. Since a detailed understanding of oscillations requires a systems biology approach, we developed a method to acquire and analyze large volumes of data on NF-κB dynamics in single cells. We measured the time evolution of the nuclear to total ratio of GFP-p65 in knock-in mouse embryonic fibroblasts using time-lapse imaging. We automatically produced a precise segmentation of nucleus and cytoplasm based on an accurate estimation of the signal and image background. Finally, we defined a set of quantifiers that describe the oscillatory dynamics, which are internally normalized and can be used to compare data recorded by different labs. Using our method, we analyzed NF-κB dynamics in over 2000 cells exposed to different concentrations of TNF- α α. We reproduced known features of the NF-κB system, such as the heterogeneity of the response in the cell population upon stimulation and we confirmed that a fraction of the responding cells does not oscillate. We also unveiled important features: the second and third oscillatory peaks were often comparable to the first one, a basal amount of nuclear NF-κB could be detected in unstimulated cells, and at any time a small fraction of unstimulated cells showed spontaneous random activation of the NF-κB system. Our work lays the ground for systematic, high-throughput, and unbiased analysis of the dynamics of transcription factors that can shuttle between the nucleus and other cell compartments.

Partial control of chaotic transients using escape times

The partial control technique allows one to keep the trajectories of a dynamical system inside a region where there is a chaotic saddle and from which nearly all the trajectories diverge. Its main advantage is that this goal is achieved even if the corrections applied to the trajectories are smaller than the action of environmental noise on the dynamics, a counterintuitive result that is obtained by using certain safe sets. Using the Henon map as a paradigm, we show here the deep relationship between the safe sets and the sets of points with different escape times, the escape time sets. Furthermore, we show that it is possible to find certain extended safe sets that can be used instead of the safe sets in the partial control technique. Numerical simulations confirm our findings and show that in some situations, the use of extended safe sets can be more advantageous.

Numerical and experimental exploration of phase control of chaos

A well-known method to suppress chaos in a periodically forced chaotic system is to add a harmonic perturbation. The phase control of chaos scheme uses the phase difference between a small added harmonic perturbation and the main driving to suppress chaos, leading the system to different periodic orbits. Using the Duffing oscillator as a paradigm, we present here an in-depth study of this technique. A thorough numerical exploration has been made focused in the important role played by the phase, from which new interesting patterns in parameter space have appeared. On the other hand, our novel experimental implementation of phase control in an electronic circuit confirms both the well-known features of this method and the new ones detected numerically. All this may help in future implementations of phase control of chaos, which is globally confirmed here to be robust and easy to implement experimentally.

PARTIAL CONTROL OF TRANSIENT CHAOS IN ELECTRONIC CIRCUITS

We present an analog circuit implementation of the novel partial control method, that is able to sustain chaotic transient dynamics. The electronic circuit simulates the dynamics of the one-dimensional slope-three tent map, for which the trajectories diverge to infinity for nearly all the initial conditions after behaving chaotically for a while. This is due to the existence of a nonattractive chaotic set: a chaotic saddle. The partial control allows one to keep the trajectories close to the chaotic saddle, even if the control applied is smaller than the effect of the applied noise, introduced into the system. Furthermore, we also show here that similar results can be implemented on a circuit that simulates a horseshoe-like map, which is a simple extension of the previous one. This encouraging result validates the theory and opens new perspectives for the application of this technique to systems with higher dimensions and continuous time dynamics.

Synchronization of uncoupled excitable systems induced by white and coloured noise

We study, both numerically and experimentally, the synchronization of uncoupled excitable systems due to a common noise. We consider two identical FitzHugh–Nagumo systems, which display both spiking and non-spiking behaviours in chaotic or periodic regimes. An electronic circuit provides a laboratory implementation of these dynamics. Synchronization is tested with both white and coloured noise, showing that coloured noise is more effective in inducing synchronization of the systems. We also study the effects on the synchronization of parameter mismatch and of the presence of intrinsic (not common) noise, and we conclude that the best performance of coloured noise is robust under these distortions.

Permutation complexity of spatiotemporal dynamics

We call permutation complexity the kind of dynamical complexity captured by any quantity or functional based on order relations, like ordinal patterns and permutation entropies. These mathematical tools have found interesting applications in time series analysis and abstract dynamical systems. In this letter we propose to extend the study of permutation complexity to spatiotemporal systems, by applying some of its tools to a time series obtained by coarse-graining the dynamics and to state vectors at fixed times, considering the latter as sequences. We show that this approach allows to quantify the complexity and to classify different types of dynamics in cellular automata and in coupled map lattices. Furthermore, we show that our analysis can be used to discriminate between different types of spatiotemporal dynamics registered in magnetoencephalograms.